What is the "initial value"?

Question

I am doing quadratic equations in math and we are doing paraballas... I came across the question , what is the "inital value". Does anyone happen to know what this means? Thank you in advance.

Answer 1

I believe this is the value of the equation when x = 0.

You might try looking up "initial value" in the index of your textbook, and finding what page this is discussed on, then check your textbook's definition versus what I've said here, just in case I've given you an answer that is different than what your class/text means by initial value.

Answer 2

In mathematics, an initial value problem is a statement of a differential equation together with specified value of the unknown function at a given point in the domain of the solution.Calling the given point t0 and the specified value y0, the initial value problem is:y'(t)=f(t,y(t)), y(t0)=y0
The problem is then to determine the function y.

This statement subsumes problems of higher order, by interpreting y as a vector. For derivatives of second or higher order, new variables (elements of the vector y) are introduced.

More generally, the unknown function y can take values on infinite dimensional spaces, such as Banach spaces or spaces of distributions.

For a large class of initial value problems, the existence and uniqueness of a solution can be demonstrated.

The Picard-Lindelöf theorem guarantees a unique solution on some interval containing t0 if f and its partial derivative are continuous on a region containing t0 and y0. The proof of this theorem proceeds by reformulating the problem as an equivalent integral equation. The integral can be considered an operator which maps one function into another, such that the solution is a fixed point of the operator. The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem.

An older proof of the Picard-Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. Such a construction is sometimes called "Picard's method" or "the method of successive approximations". This version is essentially a special case of the Banach fixed point theorem.

Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. This condition has to do with the existence of a Lyapunov function for the system.

Answer 3

Could you give an example of where this question
was posed? It's a term usually used in calculus
and I've never seen it used in connection with parabolas.